Inverse Design of Inorganic Electrides

Yunwei Zhang1, Hui Wang1, Yanchao Wang 1, Lijun Zhang1, and Yanming Ma1,*

1State Key Lab of Superhard Materials, Jilin University, Changchun 130012, China

*Address correspondence to: [email protected]

Electrides are ionic solids that consist of cationic frameworks and anionic trapped in the voids of lattices. Organic electrides exist in a large abundance, but the thermal instability at room temperature and sensitivity to moisture are bottlenecks that limit their practical uses. Known inorganic electrides are rare but appear to have high thermal and chemical stability and exhibit promising applications as -emitting materials, superior catalysts and strong reducing agents. Here, we report a developed inverse-design method that can be used to search for a large variety of inorganic electrides. Our method utilizes the intrinsic property of interstitial electron localization of electrides as the global variable function being incorporated into the swarm-intelligence based structure searches. Through screening 99 binary ionic compounds, we have designed 89 new inorganic electrides that are classified into three-, two-, and zero-dimensional species according to the way that the interstitial electrons are localized and the conductive properties of the systems. Our work reveals the rich abundance of inorganic electrides by extending them into more general forms and provides new structure types for electrides that are not thought of as before.

lectrides are ionic solids with excess electrons high-pressure electrides were predicted in various trapped at the interstitial areas of lattices, which simple elemental metals15,16,17,18,19. These elemental serve as anions1-2. The strongly or loosely electrides are unique and differ from electron-rich ionic E 20 localized electrons make electrides relate to salts with electrides seen at ambient pressure . A drawback is that F-centers or plasmas, but markedly differ from metals these high-pressure electrides are not quenchable to containing delocalized electrons3. So far, most of the ambient conditions and hardly to be of any practical synthesized electrides are organic species composed of use. alkali metals complexed with crown ethers for the Recently, a substitutional screening method21 has cations. The solvent electrons released by alkali metals been applied to search for 2D electrides. The are trapped in potential wells formed between these rhombohedral anti-CdCl2-type structure of Ca2N was complexed cations3,4,5,6. Though they are reported as used as a prototype electride structure where Ca and N low-temperature electron emitters and strong reducing were replaced with alternative metal elements (e.g., materials4,5,7, the thermal instability above -40℃ and alkali and transition metals) and non-metal elements sensitivity to air and water restrict their practical (e.g., halogen elements, O and C), respectively. On the application. one hand, the method is successful, especially when the The first inorganic electride C12A7:e- was potential electrides do crystalize in the same structure. synthesized in 2003 by removing oxygen ions from the There is the case on the prediction of Y2C electride that center of the clathrate Ca-Al-O cages of mayenite led to the actual experimental observation22. On the 8 (12CaO7·Al2O3) . It shows superior properties that are other hand, this method fails since it confines electrides thermally stable at room temperature and chemically in a certain structure type, which, however, limits our stable in resisting to water and air. Excellent understanding of the diversity of electride structures. field-emission property is evident with even lower work The predicted electrides strongly rely on the known function than carbon nanotubes9. An order of magnitude structure types, while a finding of electrides with an higher catalytic property than standard ones (such as unknown structure is impossible. 10 iron-based catalyst Fe-Al2O3-K2O ) is manifested As described above, the inorganic electrides at when loading on Ru element in the ambient conditions are rarely known. This inevitably synthesis11. limits the wide range of applications of electrides. A Dicalcium nitride (Ca2N) was recently proved to massive computation-assisted design of inorganic be a two-dimensional (2D) electride where anionic electrides is highly desirable. Here we developed an electrons are weakly localized in the 2D interspaces efficient inverse-design method that relies on the + 12 between two positively charged ionic [Ca2N] layers . intrinsic physical property of interstitial electron Ca2N was then used as an efficient electron donating localization other than any prior known electride agent in the transfer hydrogenation of alkynes and structures to search for a variety of new inorganic 13 alkenes . Monolayer Ca2N encapsulated by graphene electrides. Our method has been incorporated into our shows excellent predicted electron transport properties in-house developed CALYPSO structure prediction if compared to typical 2D electronic systems (e.g., code23,24,25 enabling automatic structure searches GaAs-AlGaAs heterojunction, LaAlO3-SrTiO3 interface through intelligent exploration of various unknown and graphene)14. structure types. Our method is reliable to identify the 12 As an important category of inorganic electrides, known electrides of Ca2N and high pressure transparent Na15 with the only given information of Eventually, we are able to construct ωvs. energy map chemical compositions. Extensive inverse-design in an effort to trace the generated structures for simulations on 99-targeted binary ionic systems were acquiring desired electrides (e.g., Fig. 1b). then performed, and 89 new electrides were successfully designed, some of which have already been synthesized, though not yet being pointed out as electrides, while others are new compounds awaiting experimental synthesis. We unraveled rich abundance of inorganic electrides by extending them into more general forms. A large variety of new prototype structures for electrides that are not thought of as before were reported.

Results and Discussion Inverse-Design Methodology. Our inverse-design method is on top of our developed swarm-intelligence based CALYPSO structure searching method that is able to intelligently explore structures with the only Figure 1. Inverse-design scheme for electrides and its benchmarks on two known electrides of Ca2N and given information on chemical composition for a transparent Na at 320 GPa. (a) Flow chart of CALYPSO compound without relying on any prior known module for searching for electrides by the inverse-design 24 structural information . In the current implementation, scheme. (b) ω vs. energy maps of Ca2N at ambient pressure the degree of interstitial electron localization other than (top left panel) and elemental at 320 GPa (bottom left the total energy was introduced and adopted as the panel). Green squares represent structures produced by global variable function. This is in good accordance CALYPSO run within 30 generations. The experimental with the fact that strong electron localization in the anti-CdCl2 and hP4 electride structures for Ca2N and sodium voids of lattice is the intrinsic physical property of are shown as red stars in left panels and depicted in top and electrides. Our structural design follows the principle of bottom right panels, respectively. inverse design where the structure searches are forced Benchmarks on Known Electrides. We have into a pursuit of a structure having a desirable benchmarked our inverse-design method onto two functionality for a given chemical system. Our previous known electride systems: Ca2N (and its variants, such inverse-design exercise has been applied into the as Sr2N and Y2C, shown in Supplementary Fig. S2) at searching of superhard materials26. ambient pressure and high-pressure electride of As a powerful technique to identify core, binding, transparent sodium at 320 GPa. With the only input and lone-pair regions in chemical systems, electron information of chemical compositions of Ca:N = 2:1, localization function (ELF)27,28,29 provides a our calculations readily reproduced the experimental semi-quantitative index for the measure of interstitial electride anti-CdCl2-type structure of Ca2N as seen in electron localization. We define the degree of interstitial Fig. 1b (top right panel) via the calculated ω vs. electron localization as the indicator to characterize the energy map (top left panel of Fig. 1b). Our calculations interstitially localized electrons. The indicator adopts also correctly reproduced the hP4 electride structure the formula of ω = Vinter/Vcry, where Vinter and Vcry are (bottom right panel of Fig. 1b) of sodium at 320 GPa as the volumes of the interstitial regions where electrons seen in the calculated ω vs. energy map of Fig 1b localize well and the simulation cell, respectively. Vinter (bottom left panel). This certifies the validity of our is critically determined from the reliable ELF method in application to design of high-pressure calculation. A detailed description on how to obtain a electrides. Note that the valence electrons of sodium at precise interstitial Vinter can be found in Supplementary 320 GPa are completely localized in the voids of hP4 Table S1 and Fig. S1. A larger ω gives a higher structure that leads to the formation of an insulating probability of the formation of electrides in a crystal. electride15. Though choices of ELF values for characterizing electron localization are dependent of specific systems, Two Design Principles for Inorganic Electrides. It is in our test, we found that ELF values larger than 0.75 known that electrides, such as [Ca Al O ]4+·(4e-) and can give a good description for electron localization for 24 28 64 [M N]+·(e-) (M = Ca, Sr and Ba), contain intrinsic most of systems. 2 excess of electrons. Existence of excess electrons in a Our inverse-design approach is outlined in Fig.1a. chemical system should be regarded as a necessary In the first generation, all structures are generated condition for stabilizing an electride and therefore has randomly with the symmetry constraints. Structures are been used as one of our design principles in searching then geometrically optimized for seeking for their local for inorganic electrides. We further prove this design ω minima in the potential surface. values of optimized principle in a model system by removing one excess structures are evaluated to rank good electride electron per formula unit of Ca2N. Upon depletion of its structures. A certain number of structures (here we excess electron Ca2N is not an electride any more chose 60% of the population size) with high-ωvalues without showing any electron localization between the are evolved into next generation by swarm-optimization + positively charged [Ca2N] layers (shown in Fig. 2a). In operation via a smart learning of personal and global below context, we define the electron-rich binary best electride structures2330. The rest 40% structures in n+ n- systems into a general form of [AxBy] ·(ne) (x, y and each generation are randomly generated to enhance n are integers; A and B are cationic and anionic structural diversity during the structure evolution. elements as electrons donor and acceptor, respectively)

Figure 2 | Tests for two design principles. (a) The electron localization function of Ca2N (left panel) on the (110)R plane parallel to the hexagonal c-axis. Upon removal of one excess electron per Ca2N, the system loses the feature of electron localization (right panel). (b) Electronegativity of elements by Pauling scale31. Metallic elements with higher electronegativity than Be are not good electron donors for binary electrides. (c) The formation probabilities (ratios of electride phases over all generated structures by CALYPSO code) of electrides in metal nitrides and Ca2X compounds are plotted in top and bottom panels, respectively. Metal and X elements are listed along x axis and formation probabilities of each compounds are shown as y axis. for searching for potential electrides. These elements with low electronegativity are preferable to electron-rich ionic systems are in contrast to those the formation of electrides. Specifically, nitrides formed conventional ionic solids (e.g., Ca3N2, YN and NaCl, by metallic elements in IA, IIA, and IIB groups and etc.) that follow the rule of electroneutrality where the aluminum have higher formation probabilities of sum of the formal charges of all constituent elements is electrides, while the probabilities are much lower for equal to zero. IVB (e.g., Zr and Hf) elements. A few electrides in Ti Electronegativity of elements is another critical and V nitrides appear, but they are energetically too principle for the design of electrides since it is a direct unfavorable to allow the experimental synthesis. Other measure of an element's ability to attract or donate metallic elements having even higher electronegativity electrons. We choose metal nitrides (La and Ac are (indicated in the bold black frame in Fig. 2b) cannot representatives of and actinides, form any electride phases. respectively) and Ca2X (X = nonmetallic elements or For the Ca2X systems, we found most of testing metallic elements with high electronegativity objects have high formation probability of electrides comparable to nonmetals, such as Ge, Sb and Bi) as beyond 50% (bottom panel in Fig. 2c), except for Ca2B two model systems to illustrate how electronegativities and Ca2-IVA compounds (e.g., Ca2C, Ca2Si and Ca2Ge). of cationic and anionic elements can influence the These latter four systems are not intrinsic electron-rich formation of electrides in certain systems. Metal compounds through intuitive calculation of the sum of + - nitrides containing excess electrons, such as [M4N] ·e the formal charges of constituent elements. Most of the + - for nitrides and [M2N] ·e for alkaline earth generated structures are therefore not electrides. metal nitrides are examined (the results on other metal Electride structures appear only when the anionic nitrides are listed in Supplementary Table S3). The elements form pairs in the structure. Once paired, the formation probability of electrides, i.e., the ratio of the resultant structure becomes an electron-rich system generated electride structures over all structures containing excess electrons satisfying the requirement produced by our inverse-design simulations, is plotted for the formation of electride. Comparing the model out in Fig. 2c. We found that only those metallic calculation results of Ca2X system with those of metal nitrides, we found that the choice of cationic elements confine our searching of electrides into those (other than the anionic elements) has a critical effect on electron-rich systems composed of cationic elements the formation of electrides. Hence, it is desirable to with low electronegativity.

Figure 3 | Inverse-design results of binary electrides. (a) Stability Map of A2B (top) and AB (bottom) (A = electrons donor, B = electrons acceptor) electrides. Black lines indicate the absence of any reasonable electride phase in that system. Squares and circles denote the predicted electrides are existing compounds and yet to be synthesized, respectively. Solid symbols indicate the energetically stable phases (have negative formation energies) with respect to the existing compounds, while open symbols for metastable phases

(have positive formation energies). (b) Type Map of A2B (top) and AB (bottom) electrides: triangles for 3D, diamonds for 0D, and crosses for 2D. Black lines indicate the same as in (a).

New Electrides by Inverse-Design Simulations. We Chemically, this explains that some of our designed conducted extensive inverse-design simulations through electrides are energetically metastable. In reality, these CALYPSO code on 99 electron-rich A2B and AB metastable electrides are experimentally synthesizable systems composed of 11 cationic A elements with low as seen in many examples on actual experimental electronegativity playing the role of electron donors and syntheses (e.g., metastable compounds denoted by open 9 anionic B elements (7 non-metallic elements and 2 squares in Fig.3a). metallic elements with high electronegativity) as According to the way the excess electrons are electron acceptors as shown in Fig. 3. Our calculations localized and the conductive properties of systems, readily reproduced the correct anti-CdCl2 structure these electrides can be classified into three categories: shared by four known Ca2N, Sr2N, Ba2N and Y2C 3D, 2D and 0D species as shown in Type Map (Fig. 3b), 1,12,22 electrides . We ruled out the formation possibility marked by different symbols. There are 52 3D, 22 2D, of electrides in nitrides and carbides dominating the and 15 0D inorganic electrides, respectively. Below, we hexagonal structure. selected three compounds of Ca2C, Be2N and LaCl as We here reported 89 newly designed electrides as the illustrative examples for 3D, 0D and 2D electrides, depicted in the Stability Map (Fig. 3a). 3D ELF maps, respectively, to discuss their structural and electronic structure information and formation energies of these properties. electrides are shown in Supplementary Table S2. It is noted that for one particular system, many electride Structures and Electronic Properties of Designed structures are generated in our simulation, however; Electrides. only the electride having the lowest-energy structure has been presented in Fig. 3a. Among these 89 3D Electrides. In 3D electrides (shown as triangles in electrides, 19 are existing compounds (square symbols Fig. 3b), excess electrons are partially localized in the in Fig. 3a), but yet to be pointed out as electrides. The cavity interstitials of the lattice. There is a subtle other 70 electrides (circle symbols in Fig. 3a) are balance between localization and delocalization of hitherto unknown compounds awaiting experimental excess electrons that contribute to the conductivity in synthesis: 17 electrides are energetically stable (i.e., three dimensions. Ca2C is a typical 3D electride that has having negative formation energies) shown as solid a body-centered tetragonal I4/mmm structure (right 2+ circles in Fig. 3a, while the other 53 are metastable (i.e., panel in Fig. 4a) consisting of cationic Ca and anionic 2- having positive formation energies) shown as open C2 . This structure is a new prototype structure of circles. It is noteworthy that our target systems contain electride that doesn't follow any known structures in the 32,33 intrinsic excess electrons and therefore go against the databases, and it can be derived from the insulating 34 electroneutrality for acquiring stable compounds. calcium dicarbide CaC2 (left panel in Fig. 4a) by replacing the dashed C2 dimers with the interstitially localized electrons. ELF results (right panel in Fig. 4b) showed that excess electrons are partially localized in anionic cavity sites that are linked each other by delocalized electrons. The calculated partial electron density for region near Fermi level (right panel in Fig. 4c) illustrates that these delocalized electrons dominate the conductivity of Ca2C. The 3D conducting behavior of excess electrons can also be inferred from the band structure of Ca2C (left panel in Fig. 4c) at Fermi level that crosses over high symmetric directions along Z-A, A-M and X-Γ.

Figure 5 | Semiconducting Be2N as a 0D electride. (a) 3D ELF of Be2N (left panel) with an isosurface value of 0.83. Six-fold and five-fold coordinated Be atoms are pointed out as inset structural units in the left panel. 2D ELF on (110) plane is plotted out in right panel to show the isolated interstitial localization regions. (b) Band structure and partial density of states (PDOS) of Be2N. The use of HSE06 functional35,36 corrects the DFT bandgap of 1.51 eV to 1.96 eV. Dashed line indicates the Fermi energy (EF).

Figure 4 | Structural and electronic properties of Ca2C and CaC2. (a) Crystal structure (left panel) and ELF (right 2D Electrides. In 2D electrides (shown as crosses in panel) on (01̅2) plane of CaC2. (b) Crystal structure (left panel) Fig. 3b), the excess electrons are confined within and ELF (right panel) on (012) plane of Ca2C. Interstitial interspaces between cationic layers and contribute to regions marked by dashed circles are where excess electrons the anisotropic conductivity of the system. Note that localize. (c) Electronic properties of Ca2C: band structure and localized electrons in 2D electrides are not evenly PDOS (left panel), and partial electron density for region near distributed and there are non-nuclear electron maxima EF (|E|<0.05 eV) on the (012) plane (right panel). at crystallographic positions. 2D electrides have commonly layered structures, however, not all of 0D Electrides. Electron localization of 0D electrides layered electrides could fall into this category. For (shown as diamonds in Fig. 3b) is geometrically similar example, though Sc2N and Y2N share the same layered to that of 3D electrides. However, excess electrons in structure with that of Ca2N, they are not regarded as 2D 0D electrides are entirely localized and do not electrides since the excess electrons are localized so contribute to the conductivity. As a result, these 0D well at the interstitial crystallographic sites and do not electrides show typically semiconducting/insulating contribute to the conductivity. behaviors, which are similar to some of known LaCl37 adopts a layered R-3m structure where high-pressure electrides, such as hP4 sodium at 320 atomic packing is arranged in a sequence of ABCBA (A, 15 GPa and semiconducting Aba2-40 lithium at 70 B and C are Cl- sheets, La3+ sheets, and anionic electron 16 GPa . layers, respectively) (Fig. 6a). Both La and Cl atoms Be2N adopts an R3m rhombohedral structure, a form a graphene-like arrangement in their respective new electride structure that is also out of the known sheets. 32,33 structure database . There are two kinds of N atoms ELF results (top panel in Fig. 6b) showed that the in Be2N: (i) in the first kind, N atoms are six-fold excess electrons are confined in the interlayer coordinated with Be atoms forming faces shared interstitial spacings between two La sheets, highlighted octahedrons along x-y plane; (ii) N atoms in the second in a white dashed circle. Three centers of the maximally kind are five-fold coordinated with Be atoms forming N localized regions are arrayed into one group that centralized hexahedrons (left panel in Fig. 5a). Each connects each other throughout the layer by delocalized hexahedron links to its neighboring hexahedrons or electrons. It is interesting to note that the conducting octahedrons by sharing one Be atom. ELF results (Fig. behaviors of the three centered excess electrons within 5a) showed that excess electrons are entirely localized one group are different. Partial electron density for at empty crystallographic positions (yellow spheres in region near Fermi level indicates that the excess Fig. 5a) in the interstitials. The localized electrons in electrons at both ends of one group (circled by a dashed areas are discrete, which leads to the semi-conductivity white line in bottom panel in Fig. 6b) are partially of the system (as shown in its band structure in Fig. 5c). localized and contribute to the conductivity of the system, however the excess electrons at the center of the group are well localized without showing any the use of our developed inverse-design method. This is conducting behavior. The dispersive band near Fermi in sharp contrast to the current situation of the rarely level highlighted in red in band structure (left panel in known inorganic electrides and the greatly limited Fig. 6c) is mainly occupied by interstitial electrons as practical application of inorganic electrides. illustrated by the density of state (right panel in Fig. 6c). Electron-rich ionic systems are chosen in our The high symmetric lines of Γ-M-K-Γ and A-L-H-A in simulations. This doesn't exclude the possibility on the the reciprocal space depicted in Fig. 6d are indicative of findings of electrides whose stoichiometry follow the 2D planes of the structure in real space where electrons principle of electroneutrality, which is out of scope in are localized. The red bands cross Fermi level only this study. We found that electrides can even be along these symmetric lines, showing a clear 2D extended into electroneutral systems once specific anisotropic conducting behavior of the system. chemical bondings are formed. For example, Ca2C has been placed into an electride system when C atoms form pairs. In view of the observed superconductivity in the heavily electron-doped mayenite38,39, we expect some of our predicted conducting electrides might also show the similar superconducting property. Through explicit calculations of electron-phonon interaction, we found that 3D electrides have potential to be superconducting though with low estimated Tc (e.g., 5 K for Ca2Bi and 4.17 K for Sc2As). The localized excess electrons can play an important role in the superconductivity of these compounds, which need a further study on the mechanism.

Methods Structure prediction. Our structure searching simulations are performed through the swarm-intelligence based CALYPSO method23,24 enabling a global minimization of energy surfaces merging ab initio total-energy calculations as implemented in the CALYPSO code. The method has been benchmarked on various known systems. Here, the degree of interstitial electron localization was

introduced as the fitness function in the search of Figure 6 | Structural and electronic properties of LaCl. (a) 3D ELF of LaCl with an isosurface value of 0.7. (b) 2D ELF electride materials. (top panel) on (1̅1̅0) plane and partial electron density for First-principles calculations. All the electronic region near EF (|E|<0.05 eV) (bottom panel) on the (1̅1̅0) structure calculations were performed using density plane. (c) Band structure and electron density of state of LaCl. functional theory within the Perdew-Burke-Ernzerhof Plotting details are the same as in Figure 5. (d) The high of generalized gradient approximation as implemented symmetry lines (in red color) in the first Brillouin zone. in the Vienna Ab initio Simulation Package (VASP)40. The projector-augmented wave (PAW)41 method was Conclusions adopted with the PAW potentials taken from the VASP library. The HSE06 hybrid functional35,36 reproduces Our developed inverse-design method benefits from the well the band gap of semiconductors, thus the HSE06 swarm-intelligence based structure search that is able to hybrid functional was applied to revise the band gaps42 generate hitherto unknown structure types, and of those semiconducting phases of zero-D electrides therefore the method is in sharp contrasted to the calculated by the PBE functional. The plane-wave substitution method relying on the known electride kinetic energy cutoffs of 800 eV and Monkhorst-Pack structure of anti-CdCl2 one. We are able to reveal a rich Brillouin zone sampling grid with the resolution of 2π variety of prototype structures of inorganic electrides. × 0.03 Å-1 were chosen to ensure that all the enthalpy 16 structural types out of 89 newly designed electrides are even found to go beyond the known structure calculations are well converged to better than 1 databases 32,33 (detailed information for new prototype meV/atom. Phonon dispersion and electron-phonon structures is listed in Supplementary Table S4). coupling calculations were performed with density functional perturbation theory using the We have extended electrides into more general Quantum-ESPRESSO43 package with kinetic energy n+ n- forms (such as [AxBy] ·(ne) for binary systems) by cutoffs of 80 Ry. 2×2×2 and 2×2×1 q-meshes in the considering the formal charges of constitute elements. first Brillouin zones were used in the electron-phonon The two choices of stoichiometry A B and AB are just 2 coupling calculations for Ca Sb and Sc As, two representative cases. In reality, there exist many 2 2 respectively. alternative stoichiometries as candidates for electrides not only in other binary compounds, but also in ternary, quaternary, and even more complex compounds that are ready for exploration. We believe the findings of electrides will become a general activity especially with References 11. Kitano, M. et al. Ammonia synthesis using a stable electride as an electron donor and 1. Walsh, A. & Scanlon, D. O. Electron excess in reversible hydrogen store. Nat. Chem. 4, 934– alkaline earth sub-nitrides: 2D electron gas or 940 (2012). 3D electride? J. Mater. Chem. C 1, 3525 (2013). 12. Lee, K., Kim, S. W., Toda, Y., Matsuishi, S. & Hosono, H. Dicalcium nitride as a 2. Dye, J. L. Physical and chemical properties of two-dimensional electride with an anionic and electrides. Annu. Rev. Phys. Chem. electron layer. Nature 494, 336–40 (2013). 38, 271–301 (1987). 13. Kim, Y. J. et al. Two-dimensional inorganic 3. Redko, M. Y., Jackson, J. E., Huang, R. H. & electride-promoted electron transfer efficiency Dye, J. L. Design and synthesis of a thermally in transfer hydrogenation of alkynes and stable organic electride. J. Am. Chem. Soc. 127, alkenes. Chem. Sci. 6, 3577–3581 (2015). 12416–12422 (2005). 14. Zhao, S., Li, Z. & Yang, J. Obtaining 4. Ichimura, A. S., Dye, J. L., Camblor, M. a. & Two-Dimensional Electron Gas in Free Space Villaescusa, L. a. Toward inorganic electrides. without Resorting to Electron Doping : An J. Am. Chem. Soc. 124, 1170–1171 (2002). Electride Based Design. J. Am. Chem. Soc. 136, 13313–13318 (2014). 5. Dye, J. L. Electrides: early examples of quantum confinement. Acc. Chem. Res. 42, 15. Ma, Y. et al. Transparent dense sodium. Nature 1564–1572 (2009). 458, 182–185 (2009).

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Author Contributions: No.11274136 and No.11534003, 2012 Changjiang Y. M. designed the research. Y. Z. performed most of Scholar of Ministry of Education. L.Z. acknowledges the calculations. Y. Z., H. W., and Y. W. coded the inverse design method into CALYPSO. Y. Z. and Y. M. funding support from the Recruitment Program of analyzed the results and wrote the manuscript. All Global Experts (the Thousand Young Talents Plan). authors commented on the manuscript. Acknowledgements Competing financial interests The authors acknowledge support from National The authors declare no competing financial interests. Natural Science Foundation of China under Grants

Supplementary Information for the paper entitled “Inverse Design of Inorganic Electrides”

Yunwei Zhang1, Hui Wang1, Yanchao Wang 1, Lijun Zhang1, and Yanming Ma1,*

1State Key Lab of Superhard Materials, Jilin University, Changchun 130012, China

The detailed description on how to get the region Vinter where interstitial electron localize well.

Table S1. Period Tables of Radii (a) and bonds (b). The interstitial region (V) is determined by excluding the ionic spheres and the covalent bonding areas between two atoms. The radii of the ionic sphere Rcut (Å) at ambient condition are given in (a). Here the Rcut may be larger than Pauling ionic radii in order to repel any anionic region near atoms and have been benchmarked from ELF maps. The covalent bonding lengths (Å) we used in our work are listed in (b). The values listed in two tables are empirical values as reference, user can reset the values in the input file of CALYPSO code. Vinter can be got from interstitial region (V) and is equal to the volume of interstitial region where electron localize well, that is, where the ELF value larger than 0.75. Here we select 0.75 to distinguish localized region from delocalized region in interstitial region (V) for all systems. However, one can change the criterion depending on different systems by themselves.

Figure S1. (a) Covalent bonding areas. The covalent bonding area of one pair of bonded atoms is a tetragonal lattice depicted in pink lines, with the lattice parameters of a = b = Rcut (refer to Table S1. (a)) and c equal to the covalent bond length (refer to Table S1. (b)) between the bonded atoms in a crystal. (b) Illustration of the Vinter in a crystal. The interstitial regions where electron localized well (Vinter)

are where the ELF value larger than 0.75 in ELF map. As an illustration, Vinter are yellow parts circled by green dashed lines in (b).

Figure S2. ω(%) vs. energy maps of Sr2N (a) and Y2C (b), respectively. Green squares are 900 structures produced by CALYPSO code with 30 generations for each system. The desirable R3m

electride structures for Sr2N and Y2C are shown as red stars.

Space Prototype Fomula Z a (Å) b (Å) c (Å) ɷ Ediff Cref Group structure

Be2C

Be2N 6 R3m Be2N 2.8065 2.8065 19.8461 3.27E-02 0.1885 Be3N2, Be(s)

Be2As 6 R3m Be2As 4.2070 4.2070 16.5828 9.42E-03 -0.0611 As(s), Be(s)

Be2Sb 6 P3m1 Be2Sb 4.2939 4.2939 7.4351 1.29E-02 0.0768 Sb(s), Be(s)

Be2Bi 6 R-3m Be2Bi 3.4688 3.4688 30.2381 1.60E-02 0.0174 Bi(s), Be(s)

Be2O 6 R3m Be2N 2.6114 2.6114 24.7554 5.66E-02 0.269 BeO, Be(s)

Be2S BeCl 4 R-3m ZrCl 3.1396 3.1396 25.7290 3.61E-02 0.508 BeCl, Be(s) BeBr

Mg2C

Mg2N 6 Cmcm Mg2N 3.2606 14.2928 4.0728 7.14E-02 0.105 Mg3N2, Mg(s)

Mg2As 6 I4/mmm La2Sb 4.1334 4.1334 14.0811 1.25E-02 0.077 Mg3As2, Mg(s)

Mg2Sb 3 Cmmm ThH2 12.9459 3.4534 3.4606 5.65E-02 0.1313 Mg3Sb2, Mg(s)

Mg2Bi 6 Immm Te2U 13.3562 6.3702 3.6953 1.24E-02 0.060175 Mg3Bi2, Mg(s)

Mg2O 6 P-3m1 Mg2O 3.0736 3.0736 10.0092 1.37E-02 0.1452 MgO, Mg(s)

Mg2S 3 P-3m1 CdI2 3.6183 3.6183 3.6183 8.28E-02 0.138 MgS, Mg(s)

MgCl 4 R-3m ZrCl 3.6387 3.6387 27.4723 5.37E-02 0.149 MgCl2, Mg(s)

MgBr 4 R-3m ZrCl 3.7788 3.7788 29.4562 5.66E-02 0.148 MgBr2, Mg(s)

Ca2C 6 P4/mmm Ca2C 3.7592 3.7592 12.1695 2.63E-02 -0.00223 CaC2, Mg(s)

Ca2N* 3 R-3m CdCl2 3.6093 3.6093 19.2631 2.78E-03 0.09942 Ca3N2, Ca(s)

Ca2As* 6 I4/mmm La2Sb 4.5992 4.5992 15.9292 4.05E-02 0.0436 Ca5As3, Ca(s) Ca2Sb* 6 I4/mmm La2Sb 4.8060 4.8060 16.8108 0.1525753 -0.0174 Ca5Sb3, Ca(s)

Ca2Bi* 6 I4/mmm La2Sb 4.8902 4.8902 17.0871 3.68E-02 -0.0194 Ca5Bi3, Ca(s)

Ca2O 6 R-3m Be2N 3.5669 3.5669 36.4600 4.59E-02 0.113 CaO, Ca(s)

Ca2S 6 R-3m Be2N 4.0084 4.0084 39.3227 8.70E-02 0.056708 CaS, Ca(s)

CaCl 4 R-3m ZrCl 4.0676 4.0676 31.1211 7.58E-02 0.09714 CaCl2, Ca(s)

CaBr 4 R-3m ZrCl 4.2287 4.2287 31.4769 6.22E-02 0.111 CaBr2, Ca(s)

Sr2C 6 P4/mmm Ca2C 4.0603 4.0603 13.0583 6.47E-03 0.04215 SrC2, Sr(s)

Sr2N* 3 R-3m CdCl2 3.8617 3.8617 20.8583 1.36E-03 -0.291 SrN2, Sr(s)

Sr2As* 6 I4/mmm La2Sb 4.8662 4.8662 16.9452 9.90E-03 0.0735 Sr5As3, Sr(s)

Sr2Sb* 6 I4/mmm La2Sb 5.0579 5.0579 17.8723 1.23E-02 0.0219 Sr5Sb3, Sr(s)

Sr2Bi* 6 I4/mmm La2Sb 5.1512 5.1512 18.1609 1.22E-02 0.0179 Sr5Bi3, Sr(s)

Sr2O 6 R-3m Be2N 3.5669 3.5669 36.4601 5.87E-02 0.12945 SrO, Sr(s)

Sr2S 6 P4/nmm HgI2 4.2936 4.2936 12.0230 3.00E-02 0.0688 SrS, Sr(s)

SrCl 6 P4/nmm PbO 4.8790 4.8790 6.7727 2.66E-02 0.264 SrCl2, Sr(s)

SrBr 6 P4/nmm SrBr 4.2159 4.2159 8.3054 2.03E-02 0.1195 SrBr2, Sr(s)

Ba2C 6 P4/mmm Ca2C 4.318 4.318 13.8603 3.16E-03 0.0387 BaC2, Ba(s)

Ba2N* 3 R-3m CdCl2 4.0573 4.0573 22.8424 3.17E-03 0.00273 Ba3N, Ba(s)

Ba2As 6 I4/mmm La2Sb 5.0932 5.0932 18.0885 6.65E-03 0.115 Ba5As3, Ba(s)

Ba2Sb* 6 I4/mmm La2Sb 5.2865 5.2865 19.0667 1.74E-03 0.02277 Ba5Sb3, Ba(s)

Ba2Bi* 6 I4/mmm La2Sb 5.3733 5.3733 19.3419 2.11E-03 0.01833 Ba5Bi3, Ba(s)

Ba2O 6 P4/nmm Cu2Sb 5.1875 5.1875 9.4619 4.34E-02 0.05275 BaO, Ba(s)

Ba2S 6 P4/nmm HgI2 4.5648 4.5648 12.7880 7.19E-03 0.068 BaS, Ba(s)

BaCl 4 R-3m CuI 5.0894 5.0894 25.4030 2.09E-02 0.1075 BaCl2, Ba(s)

BaBr 6 R-3m CuI 5.2051 5.2051 27.483 6.73E-03 0.105 BaBr2, Ba(s)

Sc2C 3 R-3m CdCl2 3.3228 3.3228 16.5998 3.48E-02 -0.0799 Sc4C3, Sc(s)

Sc2N 3 R-3m CdCl2 3.2182 3.2182 15.9749 4.58E-02 -0.01 ScN, Sc(s)

Sc2As 6 P4/nmm Cu2Sb 3.9785 3.9785 7.4978 1.73E-02 0.0045 Sc5As3, Sc(s)

Sc2Sb* 6 P4/nmm Cu2Sb 4.2110 4.2110 7.8140 1.21E-02 -0.02 Sc5Sb3, Sc(s)

Sc2Bi 6 P4/nmm Cu2Sb 4.2949 4.2949 7.9737 1.29E-02 -0.103 ScBi, Sc(s)

Sc2O 6 I41 TiN2 4.3774 4.3774 9.6834 7.91E-03 -0.0482 Sc2O3, Sc(s)

Sc2S 6 P4/nmm HgI2 3.6395 3.6395 9.2827 1.19E-02 0.02004 ScS, Sc(s)

ScCl 4 R-3m ZrCl 3.5188 3.5188 30.2238 2.31E-02 0.0084 ScCl3, Sc(s)

ScBr 4 P-3m1 ScBr 3.6156 3.6156 10.4459 1.74E-02 0.0454 ScBr3, Sc(s)

Y2C* 3 R-3m CdCl2 3.6117 3.6116 18.4237 1.85E-02 -0.0961 Y4C5, Y(s)

Y2N 3 R-3m CdCl2 3.5219 3.5219 17.6952 2.73E-02 0.0503 YN, Y(s)

Y2As 6 Immm Y2As 13.0474 3.9671 6.0464 8.19E-03 0.0948 YAs, Y(s)

Y2Sb 6 P4/nmm Cu2Sb 4.4971 4.4971 8.4669 7.95E-03 0.0223 Y5Sb3, Y(s)

Y2Bi 6 P4/nmm Cu2Sb 4.5754 4.5754 8.6188 8.40E-03 0.0113 Y5Bi3, Y(s)

Y2O 6 P42/nmc HgI2 3.7612 3.7612 10.6135 4.14E-03 0.0825 Y2O3, Y(s)

Y2S 6 P4/nmm Y2S 3.9125 3.9125 10.1780 2.72E-03 0.0109 YS, Y(s)

YCl* 4 R-3m ZrCl 3.7308 3.7308 46.1839 9.36E-03 0.071 YCl3, Y(s)

YBr* 4 P-3m1 ScBr 3.8080 3.8080 14.6172 5.67E-03 0.0626 Y2Br3, Y(s)

La2C 3 P-3m1 La2C 3.7269 3.7269 6.9088 7.63E-04 -0.03283 La2C3, La(s) La2N 3 R-3m CdCl2 3.6793 3.6793 20.3756 5.08E-03 -0.00317 LaN, La(s)

La2As 6 P63/mmc MoS2 4.2359 4.2359 13.5345 1.48E-02 0.305 La4As3, La(s)

La2Sb 6 P4/nmm Cu2Sb 4.6811 4.6811 9.1226 1.89E-03 -0.019 LaSb, La(s)

La2Bi 3 R-3m CdCl2 4.4640 4.4640 21.1735 2.16E-03 0.281 La5Bi3, La(s)

La2O 3 R-3m CdI2 3.7951 3.7951 6.2679 6.29E-03 0.1811 LaO, La(s)

La2S 6 R3m La2S 3.9462 3.9462 40.6260 7.92E-02 0.0755 LaS, La(s)

LaCl* 4 R-3m ZrCl 4.0654 4.0654 30.3248 9.73E-03 0.055 LaCl3, La(s)

LaBr* 4 R-3m ZrCl 4.1333 4.1333 29.7150 9.01E-03 0.1015 LaBr2, La(s)

Zr2C 3 R-3m CdCl2 3.3478 3.3478 16.1704 9.51E-03 -0.057 ZrC, Zr(s)

Zr2N 3 R-3m1 CdI2 3.2919 3.2919 5.2698 6.86E-03 -0.0504 ZrN, Zr(s)

Zr2As 3 I4/mmm MoSi2 3.5674 3.5674 9.5248 2.11E-02 0.354 Zr3As2, Zr(s)

Zr2Sb 4 I4/mmm MoSi2 3.7492 3.7492 9.6087 1.46E-02 0.0481 Zr5Sb3, Zr(s)

Zr2Bi 4 I4/mmm MoSi2 3.8123 3.8123 9.7705 1.43E-02 -0.0377 ZrBi, Zr(s)

Zr2O 6 Pnnm FeS2 5.1766 5.1766 3.2179 1.30E-02 -0.3018 ZrO, Zr(s)

Zr2S* 3 P-3m1 CdI2 3.5384 3.5384 5.6365 2.00E-02 -0.175 ZrS, Zr(s)

ZrCl* 4 R-3m ZrCl 3.4272 3.4272 30.3364 2.26E-02 -0.133 ZrCl3, Zr(s)

ZrBr* 4 R-3m ZrCl 3.5258 3.5258 31.5407 1.96E-02 -0.0757 ZrBr3, Zr(s)

Hf2C 3 R-3m CdCl2 3.2869 3.2869 15.7881 4.55E-03 -0.094 HfC, Hf(s)

Hf2N 3 P-3m1 CdI2 3.2333 3.2333 5.1616 1.60E-02 -0.1358 HfN, Hf(s)

Hf2As 6 Immm Te2U 3.4908 5.6141 12.1631 1.00E-02 0.0281 Hf3As2, Hf(s)

Hf2Sb 3 I4/mmm MoSi2 3.6988 3.6988 9.5574 1.75E-02 -0.00421 Hf5Sb3, Hf(s)

Hf2Bi 3 I4/mmm MoSi2 3.7547 3.7547 9.7950 1.64E-02 -0.06017 HfBi, Hf(s)

Hf2O 3 P-3m1 CdI2 3.2187 3.2187 5.1306 4.28E-02 -0.13738 HfO2, Hf(s)

Hf2S* 6 P63/mmc NbS2 3.3748 3.3748 11.7720 3.20E-02 -0.3333 HfS2, Hf(s)

HfCl* 4 R-3m ZrCl 3.3704 3.3704 29.6932 2.50E-02 -0.1578 HfCl4, Hf(s)

HfBr* 4 R-3m ZrCl 3.5067 3.5067 31.6268 2.26E-02 -0.112 HfBr4, Hf(s)

Al2C 3 P-3m1 CdI2 3.1047 3.1047 4.4460 3.64E-02 0.0709 Al4C3, Al(s)

Al2N 6 I-4m2 Al2N 3.117 3.117 17.1591 8.71E-02 0.0826 AlN, Al(s)

Al2As 6 I-4m2 Al2N 3.9138 3.9138 19.3565 5.75E-02 0.229 AlAs, Al(s)

Al2Sb 6 I-4m2 Al2N 4.2602 4.2602 19.6040 6.82E-02 0.275 AlSb, Al(s)

Al2Bi

Al2O 2 R3m Al2O 2.8912 2.8912 10.4176 1.94 E-03 0.131 Al2O3, Al(s)

Al2S 6 P-3m1 Al2S 3.3924 3.3924 12.9981 3.05E-02 0.165 Al2S3, Al(s)

AlCl 4 Pmmn AlCl 3.1735 3.5344 8.6712 3.19E-02 0.24 AlCl3, Al(s) AlBr

Table S2. Prototype structure, relaxed lattice data, degree of interstitial electron localization, and formation energy with respect to existing compounds of all designed binary electrides. a, b, and c are crystal lattice parameters of their unit cell, respectively. ɷ is the degree of interstitial electron

localization. Ediff is the formation energy with respect to the existing stable compounds (Cref). Compounds with asterisk (*) have already been synthesized.

Elements Ratio (%) Elements Ratio (%) Elements Ratio (%) Elements Ratio (%)

Li4N 83.1 Tl2N 0.0 Hf2N 29.2 Co2N 0.0

Na4N 62.2 Ge2N 0.0 V2N 0.6 Rh2N 0.0

K4N 23.1 Sn2N 0.0 Nb2N 0.6 Ir2N 0.0

Rb4N 15.6 Pb2N 0.0 Ta2N 15.6 Ni2N 0.0

Cs4N 2.3 Sb2N 0.0 Cr2N 3.3 Pd2N 0.0

Be2N 74.1 Bi2N 0.0 Mo2N 0.2 Pt2N 0.0

Mg2N 51.8 Po2N 0.0 W2N 0,2 Cu2N 0.0

Ca2N 87.7 Sc2N 87.6 Mn2N 0.0 Ag2N 0.0

Sr2N 65.1 Y2N 71.5 Tc2N 0.0 Au2N 0.0

Ba2N 37.7 La2N 20.2 Re2N 0.0 Zn2N 0.0

Al2N 87.5 Ac2N 1.8 Fe2N 1.3 Cd2N 0.0

Ga2N 0.2 Ti2N 3.5 Ru2N 0.0 Hg2N 0.0

In2N 0.0 Zr2N 13.8 Os2N 0.0

Table S3. Chemical stoichiometries of nitrides in the model system of metal nitrides and the ratios of electride structures over all structures generated by CALYPSO code (formation probability of electrides).

Prototype Space group Z Wyckoff position

Be2N R3m 6 Be1: 3a (0, 0, 5885), Be4: 3a (0, 0, 0.1679) Be7: 3a (0, 0, 4234), Be10: 3a (0, 0, 0) N13: 3a (0, 0, 0.2931), N16: 3a (0, 0, 0.0844)

Be2As R3m 6 Be1: 3a (0, 0, -0.4342), Be2: 9b (0.1648, -0.1648, -0.3145) As5: 3a (0, 0, -0.0967), As6: 3a (0, 0, -0.5794)

Be2Sb P3m1 2 Be1: 3d (0.4983, -0.0035, 0.1496), Be2: 1b (0.3333, 0.6667, 0.8994) Sb5: 1c(0.6667, 0.3333, 0.8493), Sb6: 1a(0, 0, 0.3388)

Mg2N Cmcm 4 Mg1: 4c(1, -0.0857, 0.25), Mg3: 4c(0.5, 0.2165, 0.25) N5: 4c(0.5, 0.1815, 0.75)

Mg2O P-3m1 2 Mg1: 2d(0.3333. 0.6667, 0.2546), Mg2: 1b(0, 0, 0.5) Mg3: 1a(0, 0, 0) O5: 2d(0.6667, 0.3333, 0.3790)

Ca2C P4/mmm 2 Ca1: 2f (0, 0, 0.5), Ca2: 2h (0.5, 0.5, 0.7457) Ca3: 2h (0.5, 0.5, 0.2543), Ca4: 1a (1, 1, 1) C1: 2h (0.5, 0.5, 0.0537), C2: 2h (0.5, 0.5, 0.9463) SrBr P4/nmm 2 Sr1: 2c(0, 0.5, 0.7068) Br3: 2c(0, 0,5, 0.1370) ScBr P-3m1 2 Sc1: 2d(0.3333, 0.6667, 0.8799) Br3: 2c(0, 0, 0.2879)

Y2As Immm 4 Y1: 4j(0, 0.5, 0.3007), Y2: 4g(0, 0.2601, 0) As9: 4i(0, 0, 0.6413)

Y2S P4/nmm 2 Y1: 2c(0.5, 0, 0.6035), Y3: 2c(0.5, 0, 0.1404) S5: 2c(0.5, 0, 0.8735)

La2C P-3m1 1 La1: 2d(-0.3333, -0.6667, 0.2092) C5: 1a(0, 0, 0)

La2S R3m 6 La1: 3a(0, 0, 0.2686), La2: 3a(0, 0, 0.1162), La3: 3a(0, 0, 0.6923), La4: 3a(0, 0, 0.8594) S5: 3a(0, 0, 0.4032), S6: (0, 0, 0.9813)

Al2N I-4m2 4 Al1: 4f(0.5, 0, 0.6221), Al2: 2a(0.5, 0.5, 0.5), Al5: 2d(0, 0.5, 0.75) N9: 4e(0.5, 0.5, 0.8137)

Al2O R-3m 6 Al1: 3b(0, 0, 0.5), Al2: (0, 0, 0) Al3: 6c(0, 0, 0.7610) O5: 6c(0, 0, 0.3020)

Al2S P-3m1 2 Al1: 2d(0.6667, 0.3333, 0.0767), Al2: (0, 0, 0.7548) S5: 2d(0.6667, 0.3333, 0.6641) AlCl Pmmn 2 Al1: 2a(0, 0, 0.5859) Cl3: 2b(0.5, 0, 0.7849)

Table S4. Structural information of 16 new prototype structures.